Grade 10 Quadratics
By: Ravi Seeder
Table Of Contents
-Axis Of Symmetry (x)
-Optimal Value (y)
-Transformation (Vertical Stretch/Compression)
2. Quadratic Relations In Factored Form: y=a(x-r)(x-s)
-X-intercepts and sub/isolating
-Axis of Symmetry
-What is Factoring
-Common Factoring, Factoring Perfect Squares, Factoring Different Squares, Factoring Simple and Complex Trinomials
3.Quadratic Relations in Standard form: y= ax2 + bx + c
-Axis Of Symmetry
-Completing The Square
-Factoring back Into Factored Form
- A memory of mine in quadratics.
-How these different methods/formulas become one as quadratics
-How all relates to each other.
1. Identifying Quadratic Relations in Vertex Form: y=a(x-h)2 +k
h= Your x-intercept. (In brackets is going to be opposite form so -2 is actually +2)
k= Your y-intercept.
Your 'h' can also represent your axis of symmetry.
Your 'y' is also represented to be your optimal value.
a= Your vertical stretch/compression. NOTE: Is your 'a' is less than 1 its is going to be compression and if it is more than one its going to be a stretch. Also Note that if your 'a' is a negative number, your parabola is going to be reflecting..
Max/Min: Min because your parabola is not reflecting due to 'a'
Axis of symmetry: x=2
Quadratic equation in vertex form:
Also Note: When you are isolating for 'x' y=0 and when your
-The Step Pattern.
The original step patter is 'Over 1 up 1 and Over 2 up 4'
But in quadratic relations if there is an 'a' value the step pattern changes.
The 'a' value gets multiplied with each 'up' value.
Example: y=2(x-4)2 +1
Due to the 'a' value in this relation being 2 you must multiply each 'up' with 2.
So the new step pattern becomes: Over 1 up 2 and over 2 up 8
First and Second Differences
Word Problem In Vertex Form:
Example: h=-0.2(d-10)2 + 20
2. Identifying Quadratic Relations in factored form: y=a(x-r)(x-s) (GRAPHING)
r= your first x-intercept (Remember opposite form so -2 is actually +2)
s= your second x-intercept
a= still your vertical stretch/compression. Still reflecting or directing up depending on sign.
Axis of symmetry: Number between the two x-intercepts in this case +3. (+2 & +4)
y= Plug your axis of symmetry into the regular relation as 'x' and expand to find 'y' which will than be your vertex with axis of symmetry. (x,y)
x-intercepts are: +1 and +5
vertical stretch is: -2
Axis of symmetry: +3
So therefore the vertex is (3,8)
NOTE: You always sub and isolate for variables x and y. Isolating for x, y=0 and Isolating for y, x=0
-What Is Factoring?
For an example: We use a method known as spreading the rainbow.
We would spread the rainbow we would multiply the first two numbers/variables in each bracket. ( so 'x' multiply by 'x' which gives us x2. Than Add the two outside numbers/variables in each bracket.( 4x plus 4x which is 8x) and than multiply the last numbers in each bracket( so 4x4 which is 16)
So what were left with is our factored equation of y=x2 + 8x + 16.
when you divide the numbers trying to find the GCF it ends to be 3. So GCF=3
So your new factored equation: 3(x+3)
-Factoring Perfect Squares
Example: y=(x+4)(x+4) or y=(x+4)2
So after Factoring and spreading the rainbow, the equation will be one simple equation in standard form which will be y= x2 + 8x + 16.
-Factoring Difference Of Squares
So final answer would be y= s2 - 16
-Factoring Simple Trinomial
For an example: x2 + 4x + 6
which will factor into (x )(x )
-Factoring Complex Trinomial
-Factoring By Grouping
Word Problem In Factored Form:
An example a word problem in this form and its solution would be:
3. Quadratic Relations in Standard form: y= ax2+ bx + c
NOTE: THE 3 TERMS IN YOUR STANDARD EQUATION ARE REPRESENTED AS ABC IN ORDER. So ax2 is represented as 'a' bx is represented as 'b' and c is represented as 'c'
NOTE: YOU WILL HAVE 2 FINAL ANSWERS WHICH CAN REPRESENT X-INTERCEPTS AND YOU HAVE THOSE 2 FROM DOING SUBTRACTING AND ADDING.
The quadratic formula is:
-Example of Quadratic Formual
-Completing The Square
Method of doing Completing the square.
Word Problem In Standard Form:
For an example: 'If you could factor the expression x2 + kx + 6, what are the possible values that k could have, explain?'
These types of standard form word problems would be answered like this:
4. Me & Quadratics (Reflection)
- On my way to improving in Quadratics.
For an example: You have a standard form equation, which than you can break down and simplify it into turn into a factored form equation. ALSO it works the other way around as well. You can have a factored form equation, spread the rainbow and turn that into a standard form equation.
Quadratics In Real Life
For an example, When you are playing basketball and shooting a hoop, the path of your shot can be represented by a quadratics path.
By: Ravi Seeder